Theorem ineccnvmo 38861

Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)

Assertion

Ref	Expression
ineccnvmo	⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo

Step	Hyp	Ref	Expression
1		relcnv 6095	. . 3 ⊢ Rel ◡𝐹
2		id 22	. . . 4 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
3	2	inecmo 38859	. . 3 ⊢ (Rel ◡𝐹 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥))
4	1, 3	ax-mp 5	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥)
5		brcnvg 5853	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦))
6	5	el2v 3463	. . . 4 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
7	6	rmobii 3377	. . 3 ⊢ (∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
8	7	albii 1841	. 2 ⊢ (∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
9	4, 8	bitri 277	1 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)

Syntax hints:   ↔ wb 208   ∨ wo 858  ∀wal 1560   = wceq 1562  ∀wral 3078  ∃*wrmo 3368  Vcvv 3456   ∩ cin 3905  ∅c0 4287   class class class wbr 5102  ^◡ccnv 5648  Rel wrel 5654  [cec 8678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rmo 3369  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682

This theorem is referenced by:  ineccnvmo2  38872